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Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations. (English) Zbl 1046.34037

The authors consider the boundary value problem (*) \(x^{(2m)}(t)+(-1)^{m+1}f(x(t))=0\), for \(0<t<1,\) \(x^{(2i)}(0)=x^{(2i)}(1),\) \(i=0,1,2,\dots ,m-1,\) where \(f:\mathbb{R}\rightarrow [ 0,\infty) \) is continuous. Let \(G_{m}(t,s)\) be Green’s function for problem (*). Using Krasnosel’skij fixed-point theorem, they show that if there are constants \(0<a_{1}<a_{2}<\cdots <a_{2k}\) and \(r_{1},r_{3},\dots ,r_{2k-1}\in (0,1/2)\) such that (1) \(f(z)<a_{i}/\int_{0}^{1}G_{m}(1/2,s)\,ds\) for \(z\in [ 0,a_{i}] ,\) \(i=2,4,\dots ,2k,\) and (2) \(f(z)>a_{i}/\int_{r_{i}}^{1-r_{i}}G_{m}(1/2,s)\,ds\) for \(z\in [ a_{i}A_{m}(r_{i}),a_{i}]\), \(i=1,3,\dots ,2k-1\), where \(A_{m}(r)\) are some functions, then the boundary value problem (*) has at least \(2k-1\) symmetric positive solutions. Another similar result provides an even number of symmetric positive solutions. As a conclusion, the relationships between the results in this article and some recent related work by J. Henderson and H. B. Thompson [Proc. Am. Math. Soc. 128, No. 8, 2373–2379 (2000; Zbl 0949.34016)] are given.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 0949.34016
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References:

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